区间值Vague集转化为Fuzzy集的方法探究及应用
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摘要
在现实世界中,存在着大量的不确定的,含糊的,不完全的信息。如何精确的描述这些信息是科学研究中最重要的问题。元素和集合之间的关系在Cantor集合中只有属于和不属于两种,当所谓的集合自身的概念含糊不清时,某些元素就不能被单纯地记为属于或是不属于这个集合。由此,1965年美国控制论专家L.A. Zadeh把经典集合论的概念加以扩展,Fuzzy集合说由此产生。鉴于Fuzzy集对元素的表达仅限于支持和不支持(反对)两种情形,却对不确定(犹豫)的情形表达甚微,使得Fuzzy集理论对一些模糊性的信息无法处理和表示。为此,另一个处理模糊信息的新概念被台湾的两位知名学者W. L. Gau和D.J. Buehrer于1993年提出,即Vague集说就此产生,随之一些具有模糊性且更为丰富的不确定信息都可以有Vague理论处理,并被认为是对Fuzzy集理论的推广。Vague集的提出为处理信息和研究决策提供了一个新的、且更有力的工具,从而促进了对模糊信息的研究进程。但是,Vague集理论从被提出到现在时间甚是短暂,而且其基础理论并不是很完善,本文认为在原有文献的基础上将Vague集向Fuzzy集转化是有必要的,借助于Fuzzy集理论中更加成熟的研究成果去解决仅仅应用Vague集理论知识而无法找到答案的一些问题。
然而在处理实际问题当中,一些问题的模糊性信息表现的越来越不明显,有刚刚推出的Vague集理论来处理对某个研究对象的支持和反对的程度从单个的实数值慢慢演变成某个区间值,于是本文提出的一个新的概念—区间值Vague集.本文综合现有的相关文献在解决属性值为区间值Vague集的多属性决策时给出了区间值Vague集的未知度定义,将实数性群决策的记分函数扩展到区间值Vague上。结合Vague集理论及一些Vague集向Fuzzy集转化的方法解释,对区间Vague集向Fuzzy集转化的方法进行了研究。结合区间Vague值的加权算术和加权几何集成算子,给出了区间Vague集的隶属度,进而给出了属性值为区间Vague值的一种有效的排序方法,并通过实例说明其有效性。
In the real word, there exists quite a lot of information that is uncertain, ambiguous and incomplete. How to describe the information accurately is the most important problem in the science research. There are only two kinds of relations between the elements of the cantor set and the cantor set including belonging or not belonging to. But when the concept of the set itself is ambiguous, some elements cannot be simply regarded as belonging to the set. Thus, the United States of American C y b emetics Professor L.A Z a de h popularized the concept of cantor set in1965and fuzzy set theory is put forward.Because vague set can only express the support and opposition, cannot express indeterminacy (hesitation) clearly. A lot of fuzzy information is unable to be represented and treated with Fuzzy set theory. Thus, another concept-vague set theory to deal with fuzzy information is proposed by W. L. G au and D.J.B u e h r e r, Tai Wan scholars. Later, Vague Set is able to handle a lot of information that is much more uncertain and fuzzy, which is regarded as a development of Fuzzy set. It promote the research process of fuzzy information as the Vague Sets is proposed and is also a new and more powerful tool for processing information and decision-making. But, it is very short since V ague set theory has been proposed so far, and its basic theory is not perfect. Moreover it is necessary to transform Vague Set into Fuzzy sets on the basis of the existing literatures, so by means of more research matures of Fuzzy Sets to solve the unsolved problem only by Vague Sets.
However, some fuzzy information is not more and more obvious when dealing with the practical problems. There is an object to research the degree of support and opposition with the Vague Set theory launched and exchanged from a single real value into interval valued. So the paper presents a novel concept-Vague Set. Vague Sets knowledge degree is defined in the paper to resolve the Vague Sets multiple attribute decision-making, consolidating the existing literature. That real type group making-decision sore function is extended to the interval value Vague Sets. The method based on theory of vague set and some methods for transforming Vague set into Fuzzy set are investigated, which is about interval-valued Vague set is transformed into Fuzzy set. Based on some aggregation operators, including interval-valued Vague set weighted arithmetic aggregation operator and interval-valued Vague set weighted geometric aggregation operator, belongingness degree is defined. Furthermore, a effective sorted method to interval-valued Vague sets are given, and its validity is explained by analyzing a practical examples.
然而在处理实际问题当中,一些问题的模糊性信息表现的越来越不明显,有刚刚推出的Vague集理论来处理对某个研究对象的支持和反对的程度从单个的实数值慢慢演变成某个区间值,于是本文提出的一个新的概念—区间值Vague集.本文综合现有的相关文献在解决属性值为区间值Vague集的多属性决策时给出了区间值Vague集的未知度定义,将实数性群决策的记分函数扩展到区间值Vague上。结合Vague集理论及一些Vague集向Fuzzy集转化的方法解释,对区间Vague集向Fuzzy集转化的方法进行了研究。结合区间Vague值的加权算术和加权几何集成算子,给出了区间Vague集的隶属度,进而给出了属性值为区间Vague值的一种有效的排序方法,并通过实例说明其有效性。
In the real word, there exists quite a lot of information that is uncertain, ambiguous and incomplete. How to describe the information accurately is the most important problem in the science research. There are only two kinds of relations between the elements of the cantor set and the cantor set including belonging or not belonging to. But when the concept of the set itself is ambiguous, some elements cannot be simply regarded as belonging to the set. Thus, the United States of American C y b emetics Professor L.A Z a de h popularized the concept of cantor set in1965and fuzzy set theory is put forward.Because vague set can only express the support and opposition, cannot express indeterminacy (hesitation) clearly. A lot of fuzzy information is unable to be represented and treated with Fuzzy set theory. Thus, another concept-vague set theory to deal with fuzzy information is proposed by W. L. G au and D.J.B u e h r e r, Tai Wan scholars. Later, Vague Set is able to handle a lot of information that is much more uncertain and fuzzy, which is regarded as a development of Fuzzy set. It promote the research process of fuzzy information as the Vague Sets is proposed and is also a new and more powerful tool for processing information and decision-making. But, it is very short since V ague set theory has been proposed so far, and its basic theory is not perfect. Moreover it is necessary to transform Vague Set into Fuzzy sets on the basis of the existing literatures, so by means of more research matures of Fuzzy Sets to solve the unsolved problem only by Vague Sets.
However, some fuzzy information is not more and more obvious when dealing with the practical problems. There is an object to research the degree of support and opposition with the Vague Set theory launched and exchanged from a single real value into interval valued. So the paper presents a novel concept-Vague Set. Vague Sets knowledge degree is defined in the paper to resolve the Vague Sets multiple attribute decision-making, consolidating the existing literature. That real type group making-decision sore function is extended to the interval value Vague Sets. The method based on theory of vague set and some methods for transforming Vague set into Fuzzy set are investigated, which is about interval-valued Vague set is transformed into Fuzzy set. Based on some aggregation operators, including interval-valued Vague set weighted arithmetic aggregation operator and interval-valued Vague set weighted geometric aggregation operator, belongingness degree is defined. Furthermore, a effective sorted method to interval-valued Vague sets are given, and its validity is explained by analyzing a practical examples.
引文
[1]L.A. Zadeh. Fuzzy sets [J]. Information and Control,1965,8:338-353.
[2]K. Atanassov. Intuitionistic fuzzy sets [J]. Fuzzy Sets and Systems,1986,20:87-96.
[3]W.L. Gau, DJ Buehrer. Vague sets[J]. IEEE Transactions Systems on Man and Cybernetics,1993,23(2):610-614.
[4]Bustince H, BurIllop, Vague Sets are intuitionistic fuzzy sets [J]. Fuzzy Sets and Systems 1996.79(3):403-405.
[5]Hong D H, choi C H, Multi-criteria fuzzy decision making problems based on vague sets theory [J].Fuzzy Sets and Systems.2000.114(1):103-113.
[6]Chen S M., Measure of similarity between vague sets [J].Fuzzy Sets and Systems. 1995.74(2):217-223.
[7]Hong D H, Kim C.A note on similarity measures between vague sets and elements[J].Information sciences,1999,115(14):83-96.
[8]李凡,徐章艳.vague集之间相似度量[J].软件学报,2001,12(6)：922-927.
[9]Szmidt E. Kacprzgk J. Distances between intuitionstic fuzzy sets [J]. Fuzzy Sets and Systems,2000:(1413):505-518.
[10]Li D F, Cheng C T, New similarity measures of intuitionstic fuzzy sets and applications to pattern recongnitions[J].Pattern Recongnitions Letters,2002,23[13]:221-225.
[11]Licmy Z Z,Shi P F. similarity measures of intuitionstic fuzzy sets and applications to pattern recongnitions[J]. Pattern Recongnitions Letters,2002,23[10]:221-225.
[12]张诚一,李亚东,党平安,vague集之间的相似度量[J].计算机科学,2003,30(5)：98-100.
[13]李凡,吕泽华,蔡立晶,基于Fuzzy集的vague集的模糊熵。华中科技大学学报(自然科学报),2003,31(1)：1-3.
[14]Hong D H, c hoi C H. M ultIc rite r I a fuzzy decision making problems based on vague sets theory [J], Fuzzy Sets and Systems.2000.114 (1):103-113.
[15]李凡,卢安,蔡立晶,基于Vague集的多目标模糊决策方法[J].华中科技大学学报(自然科学报),2001,29(7)：1-3.
[16]Atanassov K. Pasi G, Yager R. Intuitionstic fuzzy interpretations of multi-persion.multicrteria decision-making A.Proceedings of First interuation IEEE Symposivemon Intelligent System[C].Piscataway,N J,USA:IEEE,2002:115-119.
[17]马志峰,邢汉承,郑晓林,区间值Vague集决策系统及其规则提取方法[J].电子学报,2001,29(5)：585-589.
[18]Nilsson J, Bernhardsson B, Wittenmark B. Stochastic analysis and control of real time systems with random time delays [J]. Computer& Chemical Engineering,2004,28(8):1337-1346.
[19]Palhares R M, Peres p L D. Robust filtering with guaranteed energy Lo peak performance-An LMI approach[J].Autamatica,2000,36(6):851-858.
[20]付海东,卢正鼎,基于Vague集的多评价指标模糊决策方法[J].华中科技大学学报(自然科学报),2003,31(8)：77-79.
[21]马志峰,邢汉承,Vague集决策表中的含糊规则获取策略[J].计算机学报,2001,24(56)：382-389.
[22]李凡,近似模糊推理[M].北京：科学出版社。
[23]Gau W. Dcmiel J B. Vague Sets [J].IEEE Transaction on systems.Man and Cybernetics,1993,23 (2):610-614.
[24]李凡,卢安,蔡立晶,关于Vague集的模糊熵及其构造[J].计算机应用与软件,2002,19(2)：10-12.
[25]林志贵,刘英平,徐立中,模糊信息Vague集向模糊集转化的一种方法[J].计算机工程与应用,2004,40(9)：24-25.
[26]黄志伟,何明瑞,一种Vague集转化为Fuzzy集方法[J].计算机工程与应用,2007,43(1)：57-60.
[27]徐凤生,Vague集转化为Fuzzy集的新方法[J].计算机工程与应用,2008,44(34)：137-138.
[28]陈水利等,模糊集合理论及其应用[M].北京,科学出版社,2005.
[29]宋晓秋,模糊教学理论与方法[M].徐州,中国矿业大学出版社,2004.
[30]Zadeh L A, Fuzzy Sets [J].Information and Control,1965.8 (3):338-353.
[31]Atanassov K, Intuitionistic Fuzzy Sets:Theory and Application [M].Heidelberg Physica-Verlay.1999.
[32]Atanassov K, Gar go v G Interval-valued intuitionistic fuzzy sets[J].Fuzzy Sets and Systems,1989.31(3):343-349.
[33]徐泽水,区间模糊信息的集成算子及其在决策中的应用[J],控制与决策,2007.2.
[34]Bu rill o P, Bus tin c e H.Entropy on intuitionistic fuzzy sets and on interval-valued fuzzy sets [J].Fuzzy Sets and System,1996,78:305-316.
[35]S z m id t E, Ka c pr z y k J.Entropy on intuitionistic fuzzy sets [J].Fuzzy Sets and Systems.2001.118:467-477.
[36]林志贵等,模糊信息处理中Vague集向Fuzzy集转化的一种方法,计算机工程与应用,2004.
[37]Hwang C L, Yoonk, Multiple attributes decision making methods and applications [M]. Spriger.1981.
[38]万树平。基于区间值Vague集的多传感器信息融合的TOPSIS方法[J],传感器与微系统。2009,28(10)：64-66.
[39]刘阳丽,杨爱萍等,基于区间值Vague集的多属性决策的TOPSIS方法[J].2012.5.
[40]Chen S M, Tan J M. multi-criteria fuzzy decision-making problems based on vague set theory [J].Fuzzy Sets and Systems,1994,67(2):163-172.
[41]Hong D H, Choi C H, Multi-criteria fuzzy decision-making problems based on vague set theory [J].Fuzzy Sets and Systems,2000,114(2):103-114.
[2]K. Atanassov. Intuitionistic fuzzy sets [J]. Fuzzy Sets and Systems,1986,20:87-96.
[3]W.L. Gau, DJ Buehrer. Vague sets[J]. IEEE Transactions Systems on Man and Cybernetics,1993,23(2):610-614.
[4]Bustince H, BurIllop, Vague Sets are intuitionistic fuzzy sets [J]. Fuzzy Sets and Systems 1996.79(3):403-405.
[5]Hong D H, choi C H, Multi-criteria fuzzy decision making problems based on vague sets theory [J].Fuzzy Sets and Systems.2000.114(1):103-113.
[6]Chen S M., Measure of similarity between vague sets [J].Fuzzy Sets and Systems. 1995.74(2):217-223.
[7]Hong D H, Kim C.A note on similarity measures between vague sets and elements[J].Information sciences,1999,115(14):83-96.
[8]李凡,徐章艳.vague集之间相似度量[J].软件学报,2001,12(6)：922-927.
[9]Szmidt E. Kacprzgk J. Distances between intuitionstic fuzzy sets [J]. Fuzzy Sets and Systems,2000:(1413):505-518.
[10]Li D F, Cheng C T, New similarity measures of intuitionstic fuzzy sets and applications to pattern recongnitions[J].Pattern Recongnitions Letters,2002,23[13]:221-225.
[11]Licmy Z Z,Shi P F. similarity measures of intuitionstic fuzzy sets and applications to pattern recongnitions[J]. Pattern Recongnitions Letters,2002,23[10]:221-225.
[12]张诚一,李亚东,党平安,vague集之间的相似度量[J].计算机科学,2003,30(5)：98-100.
[13]李凡,吕泽华,蔡立晶,基于Fuzzy集的vague集的模糊熵。华中科技大学学报(自然科学报),2003,31(1)：1-3.
[14]Hong D H, c hoi C H. M ultIc rite r I a fuzzy decision making problems based on vague sets theory [J], Fuzzy Sets and Systems.2000.114 (1):103-113.
[15]李凡,卢安,蔡立晶,基于Vague集的多目标模糊决策方法[J].华中科技大学学报(自然科学报),2001,29(7)：1-3.
[16]Atanassov K. Pasi G, Yager R. Intuitionstic fuzzy interpretations of multi-persion.multicrteria decision-making A.Proceedings of First interuation IEEE Symposivemon Intelligent System[C].Piscataway,N J,USA:IEEE,2002:115-119.
[17]马志峰,邢汉承,郑晓林,区间值Vague集决策系统及其规则提取方法[J].电子学报,2001,29(5)：585-589.
[18]Nilsson J, Bernhardsson B, Wittenmark B. Stochastic analysis and control of real time systems with random time delays [J]. Computer& Chemical Engineering,2004,28(8):1337-1346.
[19]Palhares R M, Peres p L D. Robust filtering with guaranteed energy Lo peak performance-An LMI approach[J].Autamatica,2000,36(6):851-858.
[20]付海东,卢正鼎,基于Vague集的多评价指标模糊决策方法[J].华中科技大学学报(自然科学报),2003,31(8)：77-79.
[21]马志峰,邢汉承,Vague集决策表中的含糊规则获取策略[J].计算机学报,2001,24(56)：382-389.
[22]李凡,近似模糊推理[M].北京：科学出版社。
[23]Gau W. Dcmiel J B. Vague Sets [J].IEEE Transaction on systems.Man and Cybernetics,1993,23 (2):610-614.
[24]李凡,卢安,蔡立晶,关于Vague集的模糊熵及其构造[J].计算机应用与软件,2002,19(2)：10-12.
[25]林志贵,刘英平,徐立中,模糊信息Vague集向模糊集转化的一种方法[J].计算机工程与应用,2004,40(9)：24-25.
[26]黄志伟,何明瑞,一种Vague集转化为Fuzzy集方法[J].计算机工程与应用,2007,43(1)：57-60.
[27]徐凤生,Vague集转化为Fuzzy集的新方法[J].计算机工程与应用,2008,44(34)：137-138.
[28]陈水利等,模糊集合理论及其应用[M].北京,科学出版社,2005.
[29]宋晓秋,模糊教学理论与方法[M].徐州,中国矿业大学出版社,2004.
[30]Zadeh L A, Fuzzy Sets [J].Information and Control,1965.8 (3):338-353.
[31]Atanassov K, Intuitionistic Fuzzy Sets:Theory and Application [M].Heidelberg Physica-Verlay.1999.
[32]Atanassov K, Gar go v G Interval-valued intuitionistic fuzzy sets[J].Fuzzy Sets and Systems,1989.31(3):343-349.
[33]徐泽水,区间模糊信息的集成算子及其在决策中的应用[J],控制与决策,2007.2.
[34]Bu rill o P, Bus tin c e H.Entropy on intuitionistic fuzzy sets and on interval-valued fuzzy sets [J].Fuzzy Sets and System,1996,78:305-316.
[35]S z m id t E, Ka c pr z y k J.Entropy on intuitionistic fuzzy sets [J].Fuzzy Sets and Systems.2001.118:467-477.
[36]林志贵等,模糊信息处理中Vague集向Fuzzy集转化的一种方法,计算机工程与应用,2004.
[37]Hwang C L, Yoonk, Multiple attributes decision making methods and applications [M]. Spriger.1981.
[38]万树平。基于区间值Vague集的多传感器信息融合的TOPSIS方法[J],传感器与微系统。2009,28(10)：64-66.
[39]刘阳丽,杨爱萍等,基于区间值Vague集的多属性决策的TOPSIS方法[J].2012.5.
[40]Chen S M, Tan J M. multi-criteria fuzzy decision-making problems based on vague set theory [J].Fuzzy Sets and Systems,1994,67(2):163-172.
[41]Hong D H, Choi C H, Multi-criteria fuzzy decision-making problems based on vague set theory [J].Fuzzy Sets and Systems,2000,114(2):103-114.
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